Although various approaches have been proposed in the literature for channel estimation in RIS-aided systems, e.g., [1]}, [2]}, [3]}, [4]}, [5]},
the required pilot signaling overhead scales with the product of the number of RIS reflecting elements and devices, which are prohibitively large in practical scenarios, especially for the case of massive access. As a remedy, the study of joint active device separation and channel estimation with a fewer number of pilots for the RIS-aided URA scheme in 6G wireless networks is desired. To achieve the goal, the considered problem can be formulated as a novel non-standard coupled tensor decomposition problem with a sparse coupled factor, assuming no knowledge about the tensor rank (i.e., the number of active devices). To handle coupled tensor decomposition type problems, various attempts have been conducted [6]}, [7]}. For instance, [6]} proposed a coupled polyadic decomposition method, where each tensor is a sum of coupled rank-1 tensors sharing some common factors. Unfortunately, the results from [6]} and [7]} cannot be directly applied to the joint active device separation and channel estimation problem with a complicated coupled structure.
Furthermore, the number of active devices is unknown in practice, which determines the number of unknown model parameters. Besides, it is generally non-deterministic polynomial-time hard (NP-hard) to acquire the number of active devices from tensor data. To address this problem, the authors in [11]} introduced an additive
regularization term that penalizes complicated channel models to mitigate the overfitting of noise. However, this method inevitably consumes enormous computational resources and leads to a heavy computation burden. To learn the tensor rank automatically, recent advances integrate the tensor rank learning into its hyper-parameter inference steps, such that the Bayesian theory can provide a natural recipe for automatic rank determination [12]}, [13]}, [14]}, [15]}. Nevertheless, the effectiveness of applying variational Bayesian inference for solving coupled tensor decomposition is still unclear in the literature. Moreover, the coupling structure (from the RIS-aided URA scheme [16]}), the element-wise sparsity structure (from angular channel representation [17]}), and the low-rank property (from the sporadic nature of mMTC [18]}) together pose unique challenges in designing proper probabilistic model and the associated inference algorithm. In this paper, by devising novel priors to incorporate problem structures, we design a novel probabilistic model to investigate the trade-off between the expressive power of the model and the tractability of the inference algorithm. The contributions of this paper are as follows:
